Correlations of sieve weights and distributions of zeros

TL;DR

This paper explores the correlations of sieve weights and their implications for prime distribution and zeros of the Riemann zeta function, providing new bounds based on recent estimates of Kloosterman fractions.

Contribution

It introduces new bounds on prime variance in short intervals and pair correlation form factors using correlations of sieve weights and recent Kloosterman estimates.

Findings

New lower bounds on prime variance in short intervals

Bounds on the form factor for zeros of the Riemann zeta function

Results rely on Bettin--Chandee's estimates of Kloosterman fractions

Abstract

In this note we give two small results concerning the correlations of the Selberg sieve weights. We then use these estimates to derive a new (conditional) lower bound on the variance of the primes in short intervals, and also on the so-called `form factor' for the pair correlations of the zeros of the Riemann zeta function. Our bounds ultimately rely on the estimates of Bettin--Chandee for trilinear Kloosterman fractions.